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Filament stretching rheometry is a prominent experimental method to determine rheological properties in extensional flow whereby the separating plates determine the extension rate. In literature, several correction factors that can compensate for the errors introduced by the shear contribution near the plates have been introduced and validated in the linear viscoelastic regime. In this work, a systematic analysis is conducted to determine if a material-independent correction factor can be found for non-linear viscoelastic polymers. To this end, a finite element model is presented to describe the flow and resulting stresses in the filament stretching rheometer. The model incorporates non-linear viscoelasticity and a radius-based controller for the plate speed is added to mimic the typical extensional flow in filament stretching rheometry. The model is validated by comparing force simulations with analytical solutions. The effects of the end-plates on the extensional flow and resulting force measurements are investigated, and a modification of the shear correction factor is proposed for the non-linear viscoelastic flow regime. This shows good agreement with simulations performed at multiple initial aspect ratios and strain rates and is shown to be valid for a range of polymers with non-linear rheological behaviour.
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https://doi.org/10.1007/s00397-021-01299-9
ORIGINAL CONTRIBUTION
Towards a universal shear correction factor in filament stretching
rheometry
F. P. A. van Berlo1
·R. Cardinaels1
·G. W. M. Peters1
·P. D. Anderson1
Received: 5 February 2021 / Revised: 28 May 2021 / Accepted: 8 August 2021
©The Author(s) 2021
Abstract
Filament stretching rheometry is a prominent experimental method to determine rheological properties in extensional flow
whereby the separating plates determine the extension rate. In literature, several correction factors that can compensate for
the errors introduced by the shear contribution near the plates have been introduced and validated in the linear viscoelastic
regime. In this work, a systematic analysis is conducted to determine if a material-independent correction factor can be
found for non-linear viscoelastic polymers. To this end, a finite element model is presented to describe the flow and
resulting stresses in the filament stretching rheometer. The model incorporates non-linear viscoelasticity and a radius-based
controller for the plate speed is added to mimic the typical extensional flow in filament stretching rheometry. The model
is validated by comparing force simulations with analytical solutions. The effects of the end-plates on the extensional flow
and resulting force measurements are investigated, and a modification of the shear correction factor is proposed for the non-
linear viscoelastic flow regime. This shows good agreement with simulations performed at multiple initial aspect ratios and
strain rates and is shown to be valid for a range of polymers with non-linear rheological behaviour.
Keywords Filament stretching rheometer ·Shear correction factor ·Viscoelasticity ·Extensional viscosity ·Numerical
simulation
Introduction
During processing, polymers undergo complex flow histo-
ries, which in general consist of a combination of shear and
extensional flow. Depending on the process, one or the other
may dominate, for instance processes such as fiber spin-
ning, film blowing or extrusion from a nozzle are largely
dominated by extensional flow. Nevertheless, even a cur-
sory examination of textbooks on rheology (Bird et al. 1987;
Macosko 1994; Tanner 2000; Morrison 2001)showsthat
rheological characterizations are mostly performed in shear
flow, whereas characterization in extensional flow is less
common. However, this is changing rapidly in the last two
decades, probably due to the development of several com-
mercial extensional rheometers (Sentmanat 2003; Hodder
P. D. Anderson
p.d.anderson@tue.nl
1Department of Mechanical Engineering, Polymer Technology,
Eindhoven University of Technology, 5600 Eindhoven,
MB The Netherlands
and Franck 2005; Huang et al. 2016). Over the years, several
measuring techniques have been developed for this purpose.
Setups such as fiber spinning (Kase and Matsuo 1965), con-
traction flows in extrusion dies (Cogswell 1972) or opposed
jets (Fuller et al. 1987) clearly have the disadvantage of
a complex geometry, thereby entailing complicated data
analysis procedures to separate shear and extensional con-
tributions which limits the accuracy of these measurement
techniques. On the other hand, stretching a material film or
filament could directly result in biaxial or uniaxial exten-
sional flow. In case of uniaxial extension, a falling mass was
initially used to generate the stretching motion (Matta and
Tytus 1990). Later, several research groups have built varia-
tions of the so-called filament stretching rheometer (Sridhar
et al. 1991; Tirtaatmadja and Sridhar 1993; McKinley et al.
1999; Anna et al. 2001; Bach et al. 2003; Chellamuthu et al.
2011;Pepeetal.2020). In this device, a cylindrical fluid
sample is placed between two parallel disks, that are then
pulled apart by moving one plate while a load cell con-
nected to the other plate measures the generated force. The
plate speed determines the extension rate. Despite the seem-
ingly simple principle, it was soon realized that extracting
/ Published online: 13 September 2021
Rheologica Acta (2021) 60:691–709
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accurate extensional viscosity data from filament stretch-
ing experiments brings along several challenges (Meissner
et al. 1981; Tirtaatmadja and Sridhar 1993). In extensional
flow, fluid elements undergo a rapid and large deforma-
tion which requires a large travelling range of the pistons or
clamps to stretch the fluid up to a sufficiently large strain
(Meissner et al. 1981). In addition, the surface tension of the
fluid will cause necking of the filament thereby resulting in
a non-homogeneous diameter and thus non-homogeneous
extension rate throughout the filament (Kr¨
oger et al. 1992).
Finally, in case of fluids with a low to moderate viscosity
that can not be clamped, the fluid is pinned to the mov-
ing pistons, resulting in no-slip boundary conditions. Hence,
the fluid in the transition region between the no-slip bound-
ary condition and the uniaxially extending filament (with a
homogeneous inwards radial velocity) undergoes a combi-
nation of shear and extensional flow, accompanied by radial
gradients in pressure. The latter pressure contributes to the
normal force on the pistons, and thereby results in an over-
estimation of the extensional viscosity at the start of the
experiment. After a certain critical strain is reached, locally
(at the middle of the sample) a uniaxial extensional flow
develops and the extensional viscosity measured at the pis-
tons converges to the theoretical pure uniaxial extensional
viscosity (Spiegelberg et al. 1996).
To overcome the above-mentioned challenges, several
improvements of the original filament stretching rheometers
have been done. Different modifications of the pistons
were implemented to reduce the deviations of uniaxial
extension at the solid-liquid contact region, including
fixing the sample with epoxy resin (M¨unstedt 1975), using
grippers (Vinogradov et al. 1970) and designing plates that
dynamically change their diameter while the extensional
test is ongoing (Berg et al. 1994). By doing interrupted
stretch experiments, it was also noticed that the error
introduced by the non-homogeneous deformation at the end
plates depends on the initial aspect ratio. Hence, performing
a pre-stretch of the sample followed by a short waiting
period to allow relaxation of generated stresses, before
performing the actual test, allows to reduce these errors
and has become common practice in extensional rheometry
experiments (Ooi and Sridhar 1994). Thereby, the ability to
reach a sufficiently large Hencky strain becomes even more
important. Using rotary clamps or rolls to stretch the sample
is an effective means of reducing the physical length of the
apparatus and has led to the development of commercial
extensional rheology add-ons for rotational rheometers such
as the SER and EVF setups (Sentmanat 2003; Hodder and
Franck 2005). However, in these devices, the sample cross-
section is in general not circular or square, which can
lead to deviations from uniaxial extension (Nielsen et al.
2009; Hassager et al. 2010). Besides this, even though the
exponentially increasing velocity required to maintain a
constant overall extension rate can be applied on the plates,
necking of the filament results in a non-homogeneous
deformation along the filament (Spiegelberg et al. 1996;
Yao and McKinley 1998). For Newtonian fluids, the local
extension rate in the thinnest region of the filament can be
50 % larger than the applied value (Spiegelberg et al. 1996)
whereas in fluids with a complex rheology the extension
rate can vary significantly throughout the experiment (Yao
and McKinley 1998). In initial studies, trial-and-error was
used to obtain an exponential radius decrease of the mid-
radius of the filament (e.g. Tirtaatmadja and Sridhar 1993).
Moreover, a master curve approach was introduced by
Orr and Sridhar (1999) to determine the optimal velocity
profile of the plates, from which an iterative approach
was developed for polymer melts (Bach et al. 2003).
Later, several control schemes were implemented that use
feedback and feed forward control combined with in situ
measurements of the filament diameter to ensure a constant
uniaxial extension rate at the middle of the filament (Anna
et al. 1999;Mar
´
ın et al. 2013). Recently, within our group, a
further improved filament extensional stretching rheometer
(FiSER) design was realized that allows to measure the
extensional rheological properties while in situ structure
characterizations can be performed (Pepe et al. 2020).
Thereto, both pistons move at equal speed in opposite
directions relying on the underlying control mechanism
whereby the stagnation point is positioned at the center of
the filament.
Besides improvements of the design and operating
principles of the filament stretching rheometers, analytical
models and numerical simulations have allowed to derive
correction factors that can compensate for the errors
introduced by the shear contribution near the plates. Using
a lubrication analysis for small initial aspect ratios (ratio
of sample height over diameter), Spiegelberg et al. (1996)
derived a first correction factor by considering an inverse
squeeze flow problem. The correction is shown to depend
on the initial aspect ratio and instantaneous Hencky strain.
Nielsen et al. (2008) provide an alternative notation of
the same correction factor, which allows to separately
insert the strains exhibited during pre-stretch and during
the actual extension experiment. By performing numerical
simulations for Newtonian fluids, Rasmussen et al. (2010)
proposed an empirical improvement of the correction factor,
which was later rewritten in terms of pre-strain and actual
strain by Huang et al. (2012). Based on the principle
of linear viscoelasticity, the derived correction factors for
Newtonian fluids are valid for all materials, as long as
they are deformed within the linear viscoelastic deformation
range (Spiegelberg et al. 1996). Yao and McKinley (1998)
found that for small strains, the fluid deformation in a
FiSER is mainly determined by the Newtonian solvent
contribution to the stress and the filament deformation is
692 Rheol Acta (2021) 60:691–709
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comparable in both the Newtonian and viscoelastic cases.
However, at large strains elastic stresses dominate leading
to strain-hardening in the axial mid-regions of the filament.
Moreover, at larger strains the initial non-homogeneous
flow, resulting from the shear regions near the end-
plates, leads to the growth of viscoelastic stress boundary
layers near the free surface which can significantly affect
the measured transient extensional viscosity (Yao and
McKinley 1998). Despite the fact that a few authors
performed full numerical simulations of the filament
stretching process for viscoelastic fluids undergoing non-
linear deformations, the validity of the available correction
factors in this regime has not been investigated. From the
findings of Yao and McKinley (1998) it is expected that
at very small strains the available (Newtonian) correction
factors are valid. However, at moderate to large strains,
the strain hardening and viscoelastic stress boundary layers
can not be described by a Newtonian model and hence the
correction factors are not expected to be valid. Moreover,
Bach et al. (2003) state that the correction factor is less
appropriate for highly non-linear materials, for which the
shearing of the sample is located very close to the end plates.
The objective of this work is to investigate whether
a general expression for the shear correction factor for
filament stretching rheometers can be determined, which
is valid in both the linear as well as in the non-
linear deformation region. By applying fully resolved
numerical finite element simulations of the fluid flow in
the filament stretching rheometer, the extensional flow and
resulting stresses on the pistons are determined for a set
of three polymers that were chosen to exhibit distinctly
different rheological behaviour in shear and/or extension.
An artificial material with more pronounced non-linear
effects is introduced as well, to investigate the extremities of
the parameter space. To describe the rheological behaviour,
the eXtended Pom-Pom (XPP) constitutive model is used,
since it nicely captures the physics of branched and linear
polymers, as shown by Verbeeten et al. (2001,2004). The
solution of the XPP model for a pure uniaxial flow will be
used as the ideal reference case. By using this solution in
the correction factor, instead of the analytical Newtonian
solution for pure uniaxial flows, it is hypothesised that even
for non-linear viscoelastic polymers a material-independent
correction factor can be derived. The correction factor is
expected to be similar to the correction factors found by
Spiegelberg et al. (1996) and Rasmussen et al. (2010)which
only depend on the initial aspect ratio and the instantaneous
Hencky strain.
The models used to simulate the flow in the FiSER
will be explained in “Modelling”. The numerical methods
used to solve these models will be reviewed in “Numerical
method”. Subsequently in “Implementation of rheological
characterization”, the method to measure the shear contri-
butions with the numerical model is described. Thereafter,
in “Results and discussion”, results are shown from the
numerical model where first a validation of the numerical
simulations is given. Finally, the results of the numerical
measurements of the shear contribution and a definition for
the correction factor is discussed.
Modelling
To perform simulations of the extensional flow in a FiSER,
multiple models have to be used. Firstly, the initial geometry
of the sample has to be determined. Subsequently, the flow
equations have to be solved on this geometry using the
appropriate boundary conditions. To do so, the non-linear
polymers have to be modeled using a suitable constitutive
equation. For this constitutive model, both linear and non-
linear material properties are needed.
Geometry
The extensional flow is created in a FiSER, by simultane-
ously moving two pistons in opposite directions with equal
velocities. To minimize the computational cost, only a part
of the FiSER is modelled, as shown in Fig. 1. Axisymme-
tryisassumedoncurveΓ3. The shape of the domain is
determined by the radius of the piston Rp, the mid-radius
R(t) and the length L(t). In FiSER experiments, the sam-
ple is slightly compressed after loading to a length Lcto
ensure good contact with the plates. The aspect ratio of the
Fig. 1 Problem description of the viscoelastic extensional flow in
a FiSER. The flow is set up by moving the end-plates in opposite
directions with equal velocities vp. The boundaries connecting the
fluid to the upper piston and the bottom piston are denoted as Γ2
and Γ4, respectively. The free surface is denoted by Γ1and the
axisymmetric boundary is denoted as Γ3
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compressed sample is then defined as Λc=Lc/Rc, with
Rcthe radius of the sample after compression. Here, it is
assumed that the compressed sample is a perfect cylinder,
or in other words the radius after compression Rcequals the
piston radius Rp(see Table 1). In addition, a slow pre-stretch
is used in experiments to increase the initial aspect ratio to
Λ0=L0/R0,whereL0=L(0)is the length of the sam-
ple after the pre-stretch and R0=R(0)is the mid-radius
of the sample after the pre-stretch. To reduce the compu-
tation time, the simulations start after the pre-stretch. First,
the initial geometry is build, where the shape of the free
surface Γ1is a circular arc or an ellipsoidal arc. An ellip-
soidal arc is used when (RpR0)>L
0/2 and otherwise a
circular arc is used. Subsequently, the pistons are simultane-
ously moved apart to a length L(t) (with a velocity vp(t )),
in such a way that the middle of the sample is extended with
a constant strain rate. A cylindrical coordinate system will
be used throughout this paper, with components [r,θ,z].
The computational domain of the FiSER is denoted by Ω.
Note that the negative z-direction is directed in the gravita-
tion direction. Besides, a balance between surface tension,
internal stresses and gravity forces determines the shape of
the free surface Γ1during extension. The piston radius, the
compressed radius and the mid-radius after pre-stretch are
constant for all simulations done in this paper. This way, the
amount of pre-strain
εpre =2ln(Rc/R0)=2ln(Rp/R0)(1)
is per definition constant. Note that the logarithmic
definition of the strain is used (i.e. the Hencky strain ε),
because it is a consistent way of describing the strain path
of a fluid element. By choosing a length after pre-strain
L0, the corresponding length after compression Lccan be
determined from the initial mid-radius R0and plate radius
Rpusing conservation of volume. So for a combination of
R0and Rp, the choice of L0determines the compressed
aspect ratio and the initial aspect ratio. In case that Γ1is a
circular arc, where (RpR0)<L
0/2 applies, the analytical
expression for Lcis found to be:
Lc=2
Rp2AB2A2+B2asin A
B
(B+R0)+A3
3A(B+R0)2+B2,(2)
with A=L0/2andB=(L0/2)2+(RpR0)2
2(RpR0).
For an ellipsoidal arc, where (RpR0)L0/2 applies,
the following analytical expression for Lcis derived:
Lc=L0π
2exp εpre
21+
1
32exp(εpre)4expεpre
2+5.(3)
Note that for a circular arc, the expression for Lccannot
be rewritten in terms of pre-strain as for the expression of
the ellipsoidal arc. Nevertheless, the circular arcs are used
because they give better approximations of the real shape
of the free surface at larger initial aspect ratios. In Table 1,
an overview is given of the four different computational
domains used in this paper. Only for the geometry with the
smallest initial aspect ratio, Ng=4, an ellipsoidal arc is
used.
It has been verified that the ellipsoidal arc and the
circular arc used in the simulated geometries in Table 1are
realistic. For these validating simulation, the pre-stretching
is simulated using a velocity of vp=0.03 m/s. The resulting
initial geometries show good agreement with the geometries
created with circular and ellipsoidal arcs.
Tab le 1 Overview of the geometries studied
NgRc
[mm]
Lc
[mm]
ΛcCompressed
shape
R0
[mm]
L0
[mm]
Λ0Initial
shape
1 1.5 1.5 1.0 1.0 2.5 2.5
2 1.5 0.73 0.5 1.0 1.27 1.27
3 1.5 0.63 0.42 1.0 1.13 1.13
4 1.5 0.37 0.25 1.0 0.68 0.68
The dimensions of the compressed and pre-stretched state are given. Simulations start from the (pre-stretched) initial shape. A constant pre-strain
εpre =0.81 is chosen
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Flow equations
To solve the flow in the filament stretching rheometer,
the momentum and mass balance have to be solved. The
following equations are used for the balance of momentum
and the balance of mass (assuming an incompressible) fluid:
ρDu
Dt =−p+∇·τ+∇·(2ηsD)+ρgegin Ω, (4)
∇·u=0inΩ, (5)
where uis the fluid velocity, ρthe density of the material,
pthe pressure and Dthe deformation rate tensor. The extra
stress tensor is defined by τ. A relatively small viscous
component ηsis added for numerical reasons and gand
egare the magnitude and direction of gravity, respectively.
Since the XPP constitutive equation will be used, the extra
stress tensor is given by:
τ=
i
gi(ciI).(6)
Here, ciis the conformation tensor of mode iand
giis the modulus of mode i, which equals ηiiwith
ηithe polymer viscosity of mode iand λithe polymer
relaxation time of mode i. The constitutive model used for
predicting the conformation tensor is the multi-mode XPP
model (Verbeeten et al. 2004). Originally, the XPP model
was proposed for branched polymers, but it also captures
the physics of linear polymers, as shown by Verbeeten
et al. (2001) for high-density polyethylene (HDPE). The
differential equation of the XPP model is given by:
ci+2exp νitrci/31
λs,i 13
trcici
+1
λb,i 3ci
trciI=0,(7)
where
cidenotes the upper convected derivative of the
conformation tensor of mode i,λb,i denotes the relaxation
time for backbone tube orientation of mode i,λs,i denotes
the backbone stretch relaxation time of mode iand the
parameter νidepends on the number of arms of the molecule
qifollowing νi=2/qi(Verbeeten et al. 2004). Isothermal
experiments are considered, wherein the temperature is
constant everywhere.
Boundary conditions
The fluid is at rest at the start of the simulation (t=0). The
initial geometry of the filament is given in Fig. 1.Onthe
driving pistons, the velocity is prescribed as:
ur=0onΓ2and Γ4,
uz=−vpon Γ2,
uz=vpon Γ4.
On the symmetry line, the following applies:
ur=0, on Γ3.
The remaining boundary is a free surface:
u·n=0, on Γ1,
where nis the outward pointing normal vector on the
surface S. On this boundary, the surface tension is applied
using a Neumann boundary condition
(pI+2ηsD+τ)·n=∇
s·(ˆγ(Inn)),onΓ1,
with sthe surface gradient operator. A constant surface
tension, ˆγis assumed.
Materials
The first material used in this study is an isotactic
polypropylene (iPP) homopolymer (Borealis HD601CF,
Mw=365 kg/mol, and Mn=68 kg/mol), which is
characterized and examined in other studies (Housmans
et al. 2009; van Erp et al. 2013; Roozemond et al. 2015;
Grosso et al. 2019). The second material that is used is
a metallocene linear low-density polyethylene (LLDPE,
ExxonMobil, Mw=94 kg/mol, and Mn=24 kg/mol),
and the material is characterized in the work of van
Drongelen et al. (2015). The rheology of both materials
is fitted with the XPP constitutive model given in Eq. 7.
The corresponding relaxation spectra and XPP parameters
are given in Tables 2and 3. In case of the iPP, two
different sets of XPP parameters are reported in literature.
The first set is given by Roozemond et al. (2015). Grosso
et al. (2019) modified these XPP parameters because of
molecular considerations. The behaviour of both the iPPs
and of the LLDPE under shear and extension is shown in
Fig. 2. These materials and their characterization are chosen
Tab le 2 Viscoelastic model parameters of the XPP model for iPP at a
reference temperature of T=220 °C
Roozemond et al. Grosso et al.
i[Pa s] λb,i [s] λs,i [s] νi[-] λs,i [s] νi[-]
1 30.00 5·1052·10526.0·1060.05
2 130.76 0.0014 4·10426.7·1060.05
3 303.60 0.011 0.0027 2 5.2 ·1050.05
4 480.00 0.060 0.015 2 2.9 ·1040.05
5 377.00 0.29 0.073 2 0.0014 0.05
6 183.70 1.67 0.42 0.25 0.0080 0.05
7 46.00 11.5 2.21 0.17 0.055 0.02
Columns 4 and 5 give the spectra used by Roozemond et al. (2015),
while columns 6 and 7 provide the values derived by using molecular
considerations by Grosso et al. (2019)
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Tab le 3 Viscoelastic model parameters of the XPP model for LLDPE (van Drongelen et al. 2015) and the arbitrary non-linear material at a
reference temperature of T=160 °C
LLDPE Van Drongelen et al. Arbitrary non-linear material
i[Pa s] λb,i [s] λs,i [s] νi[-] λb,i [s] λs,i [s] νi[-]
1 392.42 7.7·1041·10622.7·1042·1042.66
2 1196.9 0.00581 1·10615.8·1042·1042.66
3 2702.0 0.0356 1·1060.66 0.00356 2·1042.66
4 4222.2 0.227 0.0210 0.66 0.0227 0.21 2.66
5 8522.1 1.53 0.0441 0.65 0.153 0.44 2.65
6 13246 8.95 2.29 0.65 0.895 22.9 2.65
7 15179 55.4 16.2 0.55 5.54 162 2.55
because of the shear thinning behaviour, the non-linear
extensional behaviour, the difference in viscosities between
iPP and LLDPE (but similar shear viscosity of the iPPs) and
the qualitative difference of the viscosity as a function of
extension and shear rate.
From Fig. 2it can be concluded that the non-linear
behavior is completely different within this set of materials.
The uniaxial extensional viscosity can be compared with
the linear viscoelastic envelope (LVE). For the linear
viscoelastic envelope, the extensional viscosity is found as
Fig. 2 Shear and extensional viscosity as a function of the shear and
extension rate given by the XPP model. Three different materials are
used: the iPP1of Roozemond et al. (2015), the iPP2of Grosso et al.
(2019) and the LLDPE of van Drongelen et al. (2015)(T=160
°C). The linear viscoelastic prediction (LVE) for each material is also
provided
follows (Bird 1976; Zhang et al. 2012):
¯η+
LVE (t ) =3ηs+
N
i=1
ηi1exp t
λb,i .(8)
Only at very small strain rates, the non-linear XPP
model equals the linear viscoelastic envelop. Therefore, it
is expected that the non-linear behaviour of both materials
will significantly affect the flow in the FiSER.
The rheological data at selected temperature can be
superposed into a master curve at an arbitrary reference
temperature by employing a temperature-dependent factor.
For the rheological data given in Fig. 2, an Arrhenius-type
relation is used (Morrison 2001):
aT=exp Ea
R1
T1
Tref ,(9)
with aTthe temperature shift factor, Tthe absolute
temperature, Tref a reference temperature, R=8.314
[J/(K mol)] the universal gas constant and Eathe activation
energy for flow. In Table 4, the activation energies and
reference temperatures of the three materials are reported.
Herein, also the density and surface tension are given.
Numerical method
The finite element method is used to solve the filament
stretching flow problem. The initial geometry is build with
Gmsh (Geuzaine and Remacle 2009). For the interpolation
Tab le 4 Material and rheological parameters (Roozemond et al. 2015;
van Drongelen et al. 2015)
Material ρ
[kg/m3]ˆγ
[mN/m]
Ea
[kJ/mol]
Tref
[K]
iPP 800 30.2 40.0 493
LLDPE760 28.8 47.7 433
*Also used for the arbitrary non-linear material
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of the velocity and pressure, isoparametric, triangular
P2P1(Taylor-Hood) elements are used, whereas for the
conformation, triangular P1elements are used. In order to
solve the constitutive equation (XPP), the log-conformation
representation (Hulsen et al. 2005), streamline-upwind
Petrov-Galerkin (SUPG) (Brooks and Hughes 1982)and
DEVSS-G (Bogaerds et al. 2002) are used for stability.
Below the most relevant details of the numerical procedure
are outlined and more details are available in van Berlo et al.
(2020).
Mesh movement
The boundary which describes the free surface will move in
time and also the boundaries connected to the pistons move.
Therefore, it is necessary to track these boundaries and
update the mesh. The position change of the free surface is
determined from the velocity at this boundary (Lagrangian
based). The velocity of the free boundary is defined as:
dxΓ1
dt=u1), (10)
where xΓ1is the position of the free boundary and u1)
is the velocity at the free boundary Γ1. The movement of
the mesh has to be compensated. To do so, the Arbitrary
Lagrangian Eulerian (ALE) formulation is used (Hirt et al.
1974). This implies that all convective terms have to be
replaced with:
D( )
Dt =∂( )
∂t ζ+(uum)·∇(), (11)
where ∂( )/∂t|ζdenotes the time derivative at a fixed grid
point and umis the velocity of the mesh.
Weak formulation
The weak form of the problem concerning the momentum
balance and mass balance, including the boundary condi-
tions, can be formulated as follows: find u,pand ssuch
that:
(v)T, ν uGT+(Dv,2ηsD+τ)(12)
(∇·v,p) =(v,∇·(ˆγ(Inn)))Γ+(vb), (13)
H,−∇u+GT=0,(14)
(q, ∇·u)=0,(15)
d+τ(uum)·∇d,Ds
Dt g(G,s)=0,(16)
for all admissible test functions v,H,qand d. Furthermore,
Dv=(v+(v)T)/2. In this formulation a Neumann
boundary condition is used for the surface tension and
gravity is introduced via the body force b=geg.For
SUPG and DEVSS-G stabilization, the parameters τand ν
are introduced, respectively. The conformation tensor can
be derived from the log-conformation representation: s=
log c. More information on log-conformation stabilization
and the function gcan be found in Hulsen et al. (2005).
A projected velocity gradient Gis introduced with the
DEVSS-G method. Information about the discretization and
this stabilization method is provided in Bogaerds et al.
(2002).
Time integration
The system of equations is solved sequentially per time
step. To integrate in time, a (semi-implicit) backward Euler
scheme is employed for the first time step and a second-
order backward differencing scheme (semi-implicit Gear
(D‘Avino et al. 2012)) is employed for all subsequent time
steps. The system of equations on the moving domain Ωis
solved using the following steps:
First, the velocity and conformation are predicted from
previous time steps using
ˆ
un+1=un(17)
ˆ
cn+1=cn(18)
for the first time step, and
ˆ
un+1=2unun1(19)
ˆ
cn+1=2cncn1(20)
for all subsequent time steps. Here, ˆ
un+1is the prediction
of the velocity and ˆ
cn+1the prediction of the conformation
tensor for time tn+1,andun,un1,cnand cn1are
the velocities, and conformation tensors at time tnand
tn1, respectively. Next, the velocity and pressure, un+1
and pn+1, are solved. The method of D‘Avino et al.
(2012) for decoupling the momentum balance from the
constitutive equation is applied, using DEVSS-G for
stability (Bogaerds et al. 2002). After solving for the new
velocities and pressures, the actual conformation tensor
cn+1is found using a second-order, semi-implicit Gear
scheme with conformation predictions from Eq. 7,where
the log-conformation representation and SUPG are used for
stability. Subsequently, the boundary positions are updated,
where the movement of the boundary is Lagrangian based
according to Eq. 10, using a backward Euler scheme
xn+1
Γ=xn
Γ+un+1(Γ )Δt. (21)
The mesh velocities can now be obtained by numerically
differentiating the mesh coordinates. The mesh velocity
is obtained in each node using a first-order backward
differencing scheme (Jaensson et al. 2015):
un+1
m=xn+1
mxn
m
Δt ,(22)
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where un+1
mis the mesh velocity at time tn+1,andxn+1
m
and xn
mare the mesh coordinates at time tn+1and tn,
respectively.
Remeshing and projection
As the mesh is deformed in time, because both pistons
move apart, elements may become too deformed to yield
accurate solutions. A new mesh is generated by tracking the
deformation of each element and determining the change in
area and the change in aspect ratio of these elements. The
change in area, fe
1, and aspect ratio, fe
2, of each element are
defined as (Jaensson et al. 2015):
fe
1=|log(Ae/Ae
0)|(23)
fe
2=|log(Se/Se
0)|,(24)
with Aethe element area, Ae
0the element area of the
undeformed mesh and Se=(Le
max)2/Ae
0the aspect ratio,
where Le
max is the maximum length of the sides of an
element. Remeshing is invoked if either fe
1>0.6 or fe
2>
0.6, which coincides with a change in area or aspect ratio by
a factor 1.8. Remeshing implies that a new mesh, covering
the same domain as the old one, is generated using Gmsh
(Geuzaine and Remacle 2009). To ensure accuracy of the
solution, the free surface consists of at least 5 elements
in the radial direction. This condition is only used in case
the mid-radius becomes very small, where as a result the
amount of elements drastically increases. Initially about 14
elements in the radial direction are used. After remeshing,
the solution on the old mesh is projected onto the new
one. The projection problem is solved to obtain the solution
variables on the new mesh. The projection is done similar as
was done by Jaensson et al. (2015).
Implementation of rheological
characterization
The desired material function to be measured with a
filament stretching rheometer is the transient extensional
viscosity, η+, as a function of time and strain rate. This
material function is defined as the average stress difference
divided by the strain rate:
η+(t, ˙ε) =σzz(t ) σrr (t )
˙ε,(25)
where σis the total-stress tensor for a fluid undergoing
homogeneous uniaxial extensional flow σzz and σrr are the
total stress in radial and axial direction. In the simulations,
the total-stress tensor follows from the extra stress tensor
and pressure according to:
σ=τpI. (26)
As time increases, the transient extensional viscosity may
reach a steady-state value, η. The (steady-state) extensional
viscosity is a material property of the fluid and is a function
of only the strain rate (and temperature). To measure correct
values for the extensional viscosity in a filament stretching
rheometer, the stress is measured via the normal force on
the bottom piston and a constant extension rate should be
ensured at the mid-filament point. The implementations of
both aspects are detailed below.
Forces
In our home-build FiSER, the force transducer is positioned
at the bottom piston (Pepe et al. 2020). This means that
during the simulations the force is measured at boundary
Γ2. The measurement of the force is done by calculating
the reaction forces. The reaction force (or the internal force)
can be found via the residual, once the solution of the
momentum and mass balance problem is known (i.e. the
displacement vector).
Since the boundary at the piston consists only of Dirichlet
nodes, the total reaction force, acting on the piston, can
be found by summation of the force contributions in the
z-direction:
Fp=(Freac,z)Γ2. (27)
Here, Fpis the force acting on the piston and Freac,z is the
z-component of the reaction force vector. In addition to the
viscoelastic material stresses, surface tension and gravity
also contribute to the measured force Fp. Therefore, these
contributions are subtracted when calculating the measured
extensional viscosity (Tirtaatmadja and Sridhar 1993; Szabo
1997):
η+
R=Fp(t)/ πR2(t)−ˆγ/R(t)ρgV /(2πR2(t))
˙ε,
(28)
with Vthe volume of the polymer sample. The subscript
Rindicates “real” in the measured viscosity, because the
force contribution of the surface tension ˆγand gravity are
subtracted. In this equation, the effect of inertial forces
is neglected. For polymer melts, this inertia term can
usually be neglected in the determination of the extensional
viscosity (Tirtaatmadja and Sridhar 1993). This is also the
case in this study, because the Reynolds number of the flow
problem is much smaller than one. The extensional viscosity
measured with the FiSER simulation, as given in Eq. 28,is
used to determine the shear correction factor, fshear.Inthis
paper, the following definition for the shear correction factor
is used:
fshear =¯η+
XPP
¯η+
R
. (29)
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Here, ¯η+
XPP is determined by solving the XPP constitutive
equation (see Eq. 7) for a pure uniaxial extensional flow
starting from rest. This definition is different from other
work in literature about the shear correction factor. Therein,
the pure uniaxial extensional viscosity was calculated using
(analytical) solutions of a linear viscous model (Spiegelberg
et al. 1996; Rasmussen et al. 2010). This modified definition
is needed to study the validity of using a shear correction
factor under non-linear conditions.
Controller
In uniaxial extension, the mid-radius of the sample has to
decrease exponentially in time:
R=R0exp 1
2ε,(30)
with εthe mid-radius-based Hencky strain. In case of
a perfectly cylindrical sample, the pistons should be
moved apart so that the gap between the pistons increases
exponentially in time. In reality, however, the lack of
uniformity of the flow along the axis of extension does not
allow to a priori determine the required piston movement.
Therefore, a strain-based controller is developed by Mar´
ın
et al. (2013), which adjusts the piston velocity so that
the mid-radius of the sample decreases exponentially in
time. In the proposed scheme, the motion of the pistons is
commanded by controlling their velocity via the Hencky
strain. In the feed-back loop, the strain is corrected by
comparing the actual radius with the ideal radius of the
middle of the filament, Rideal. To do so, the radius of the
middle of the filament is measured with a laser and is used
as an input in the next time step. In the simulations, the
minimum radius of the free surface is determined per time
step iand is defined as the measured radius R(i).The
equation for the controller of the length-based Hencky strain
isgivenbyMar
´
ın et al. (2013):
εz(i +1)=εz(i) +Kp[δε(i) δε(i 1)]
+KiΔt[δε(i)]+Δεff
z(i). (31)
The corresponding filament length and piston velocity
can be calculated with the following relations:
L(i +1)=L0expz(i +1)). (32)
vp(i +1)=L(i +1)L(i)
Δt . (33)
In control language, εzis the actuated variable and εthe
controlled variable. In Eq. 31, the error δε is calculated as
follows:
δε(i) =εideal(i) ε(i) =2lnR(i)
Rideal(i ) (34)
where ε(i) is the measured mid-radius-based Hencky strain
at time step i, which follows from the measured radius using
Eq. 30.Rideal is the ideal radius for pure uniaxial flow. The
ideal radius also follows from Eq. 30 using that ε(t) εt.
The actual radius of the sample is measured with a laser
and is defined as R(=Rmeas). In the home-build FiSER,
the time step of the laser is Δt =200 μsandthePI
gains are Kp=0andKi=2.5 s1(Pepe et al. 2020).
However, in the simulations KiΔt =0.08 is used. At the
largest strain rate of 10 s1, a time step of 20 μsisused
and hence Ki=4000 s1(Kp=0). The time step and
integral gain depends linearly on the strain rate, with lower
limits of Δt =200 μsandKi=400 s1, respectively.
For the controller and the simulation, the same time step
is used. Also a feed-forward contribution Δεff
zis present
in the controller of the home-build FiSER. However, in the
simulations, this feed-forward contribution is superfluous
and therefore set to zero (Δεff
z=0).
Results and discussion
First, a validation of the simulations is given. Here, the
controlled radius is validated and the simulated forces
are compared with measurements from literature. This is
followed by a convergence study. Subsequently, strain rate
distributions over the radius of the sample are shown.
Finally, the shear correction factor is discussed, where the
effects of initial aspect ratio and strain rate on the shear
correction factor are shown.
Validation
One of the requirements for a pure uniaxial flow is that
the mid-radius of the sample decreases exponentially with
strain. As mentioned before, a controller is used to ensure
this exponential decrease (see “Controller”). In Fig. 3,the
mid-radius of the filament is given as a function of the
Hencky strain. Here, the simulated radius immediately (after
Fig. 3 Mid-radius measurements with the home-build FiSER at a
strain rate of ˙ε=0.71 s1and temperature of 150 °C (Pepe et al.
2020). The sample dimensions are: Rc=4 mm, Lc=1.14 mm,
R0=1.68 mm and L0=3.64 mm. The solid line represents the
simulation of the FiSER with the XPP model and the dashed line is the
ideal mid-radius following from Eq. 30
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one time step of ε=0.015) follows the ideal exponential
profile. The slopes of the measured and ideal radius are
identical. This means that the strain-based controller of
the simulations is fast and works properly. In the FiSER
experiments, the controller is slower, because a lower value
of the integral gain (Ki=2.5 s1)isused.
To validate the simulations, the force results are
compared with simulated force results from the work of
Kolte et al. (1997). Herein, the flow of polyisobutylene
(PIB) in a FiSER is simulated using a multi-mode
viscoelastic Oldroyd-B constitutive equation. Tirtaatmadja
and Sridhar (1993) performed FiSER measurements with
an exponential decreasing mid-radius and measured the
force at the plate. A comparison of all simulated and
experimentally determined forces is given in Fig. 4.Forthe
setup used by Tirtaatmadja and Sridhar (1993) the early
response of the fluids is not only masked by the shear
correction factors but also by the dynamics of the drive train.
The drive has to accelerate from zero velocity to a large
velocity instantaneously (L0˙ε), which takes approximately
0.1 seconds in this experiment. Note that for more recent
filament stretching rheometers, drives accelerate faster
and controllers can follow the desired exponential radius
evolution from very small strains (see Fig. 3). Hence, the
effects of acceleration are not studied in this paper. In Fig. 4,
it can also be seen that our simulated force matches the
simulated force of Kolte et al. (1997). For the used Oldroyd-
B model an analytical solution for the purely uniaxial
extensional viscosity can be found as (Kolte et al. 1997):
¯η+
Old-B (t, ˙ε) =3ηs+
N
i=1
ηi3
(12Dei)(1+Dei)
2exp1
Dei2˙εt
12Dei
exp 1
Dei+1˙εt
1+Dei
,(35)
Simulation
Kolte et al. (simulation)
Tirtaadmadja et al. (experiment)
Pure uniaxial solution (Oldroyd-B)
Fig. 4 Force as a function of the Hencky strain of a sample extended
with a strain rate of ˙ε=2.0 s1using a four mode Oldroyd-B model.
The solvent viscosity, polymer viscosities and relaxation times of the
PIB used are ηs=12.4 Pa s, η=[1.69 2.56 2.53 1.85]Pa s and
λ=[4.20 1.12 0.167 0.0149]s. The surface tension is neglected and
the sample dimensions are Lc=L0=1.5 mm and Rc=R0=1.5
mm
with Deiελithe Deborah number for the i’th
mode and Nthe number of modes. With this extensional
viscosity, the analytical (pure uniaxial) force is found by
using Eq. 28, whereby that gravity and surface tension
are not contributing. This pure uniaxial solution is also
showninFig.4. By comparing the simulations and the
pure uniaxial solution, it can be concluded that at the start
of the simulations, the flow is not purely uniaxial, since
the force overshoot is larger in the simulation of an actual
filament stretching experiment. This difference between the
pure uniaxial force and the experimental force is a result
of shear contributions near the plates (Spiegelberg et al.
1996). To simulate the flow in the FiSER with the XPP
model, the same governing equations as for the Oldroyd-
B model are used. Therefore, only the constitutive equation
is changed when using the XPP model. In the work of
Baltussen et al. (2010) it can be seen that the XPP model is
correctly implemented in the in-house developed software
package TFEM.
Mesh and time step convergence
To study the mesh convergence of the system, simulations
are performed on five different meshes given in Table 5.
Here, nini =Rp/hini is the number of elements at the piston
boundaries, with hini the initial element size. At the start of
the experiment, the initial element size is used in the entire
domain. For the convergence study, remeshing is included.
During remeshing, the mesh at the middle of the filament
is refined, while the amount of elements at the boundary
is kept the same as the initial element size. Remeshing is
performed such that at least nmid =5 elements are present
over the mid-radius R(hmid R/nmid), even at large strains
where the radius becomes very small. Simulations up to a
strain of ε=1.5 are performed, with a time step of Δt =
105s, a strain rate of ˙ε=10 s1and a temperature of
T0=200 °C. The sample dimensions are Rp=4, R0=3
and L0=6 mm. In Fig. 5, the simulated forces are shown
for the different meshes. At a strain of 1.5, the simulated
force of the reference solution is close to the pure uniaxial
force determined with the XPP model (as expected). Also a
Tab le 5 Mesh resolution of different surface meshes in the conver-
gence study
Mesh nini hini [mm] nmid
M1 2 2.0 5
M1 4 1.0 5
M3 6 0.67 5
M4 10 0.4 5
M518 0.22 5
The star symbol indicates the reference mesh
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Fig. 5 Simulated forces and the pure uniaxial solution calculated with
the XPP model. The reference force curve has an initial element size
of h
ini =0.22 mm
convergence with increasing mesh resolution of these forces
can be seen at the maximum force (ε=0.37).
In order to investigate this convergence in a more
quantitative manner, the following error is defined:
F(ε) =|F(ε)F(ε)|
F(ε) ,(36)
with Fthe simulated force and Fthe simulated force for
the reference mesh M5as indicated in Table 5. The error
is calculated for the different meshes at a strain of ε=0.37
and ε=1.5, as shown in Fig. 6a. The convergence at a
strain of 0.37 seems to be cubic. This rate of convergence
is expected based on the order of interpolation of the
elements (quadratic elements). For a strain of 1.5, the
convergence seems to be quadratic. This is lower than the
expected convergence rate. One of the possible reasons is
that remeshing changes the convergence rate. Also other
numerical errors can affect the convergence rate. In this
work, the M4 mesh is used to simulate the flow in the
FiSER. The error of this mesh compared to the reference
mesh is smaller than 104at a strain of 1.5.
A suitable time step has to be chosen for the simulations.
Therefore, a time dependency study is performed on the
M4 mesh. The same conditions are used as for the mesh
dependency study. There is a limit on how large the time
step size can be, because of the controller (Δtmax =2·
104). Besides, at higher strain rates, the time step is
linearly decreased to maintain stability of the simulation.
Equal time steps for the controller and the simulation are
used. The time dependency is studied in a quantitative
manner by using Eq. 36. The reference time step is taken
as Δt
1=1·107s. In Fig. 6b, the result of the
convergence with the time step is shown. For both strains,
the convergence seems to be linear. This figure also shows
that the errors are quite small compared to the error in the
mesh convergence, indicating that it is only important that
the time step is smaller than the time step limit due to the
controller. The convergence rate is lower than the expected
one, since a second-order time stepping scheme is used.
This is possibly caused by the remeshing and the first-order
mesh movement.
Shear correction factor
In this section, FiSER simulations are investigated for
multiple geometries and materials. The simulation domains
used in this study are given in Table 1. For geometry 2
(using Λc=0.5) at a strain rate ˙εof 10 s1the simulation
results at different Hencky strains are given in Fig. 7.
Herein, the shape evolution of three polymer melts (iPP1
(Roozemond et al. 2015), iPP2(Grosso et al. 2019)and
LLDPE (van Drongelen et al. 2015)) and of an arbitrary
non-linear material are shown. From Fig. 7it is clear that at
small strains (ε1), the shapes of the free surface of the
polymer melts match. But for Hencky strains larger than 1,
the controller changes the velocity of the pistons to control
the mid-radius. Therefore, the length of the samples and
their free surface shapes are different, but the mid-radius is
exactly the same.
In Fig. 8, the extensional viscosity which follows from
these FiSER simulations is shown. By comparing the
simulations with the pure uniaxial solutions it can be
concluded that, although the strain rate in the middle of
the sample is constant, the flow is not purely uniaxial at
Fig. 6 Mesh and time step convergence at a strain of 0.37 and 1.5. In
(a), a time step of Δt =105s, a strain rate of ˙ε=10 s1and a
temperature of T0=200 °C are used. In (b), a M4 mesh is used with
˙ε=10 s1and T0=200 °C
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Fig. 7 Simulated boundaries of the iPPs, LLDPE and the arbitrary
non-linear material at strains of ε=0.5, ε=1, ε=2andε=3. The
second initial geometry is used with Λc=0.5. The mesh evolution is
shown for iPP2of Grosso et al. (2019) and the filaments are extended
with a strain rate of ˙ε=10 s1at T=160 °C
small strains. Therefore, the real extensional viscosity of the
simulation needs to be corrected (as explained in “Forces”).
At large strains (ε>2), the simulations converge towards
the pure uniaxial solution. This implies that there is no
correction needed at large strains. In this case, the flow at
the middle of the filament behaves perfectly uniaxial (like
a perfect cylinder). The deviation from a pure uniaxial flow
has already been shown in multiple studies (Spiegelberg
et al. 1996; Nielsen et al. 2008; Rasmussen et al. 2010;
Huang et al. 2012). In the work of Spiegelberg et al. (1996)
it is assumed that the fluid sample is viscous and has an
initial aspect ratio smaller than one. Therefore, a lubrication
approach can be used to describe the initial deformation
of the fluid sample. Since this fluid sample is confined
between two cylinders, the problem resembles a squeeze
flow problem (with reversed direction of motion). Based
on this theory, an analytical expression for the correction
factor has been derived. Nielsen et al. (2008) have rewritten
this analytical expression in terms of strain and pre-strain.
The analytical correction factor which follows from this is
(Spiegelberg et al. 1996; Nielsen et al. 2008):
fshear =1+exp(7(ε(t ) +εpre)/3)
3Λ2
c1
. (37)
Here, t=0 is defined after pre-stretch and therefore R0
is the mid-radius of the polymer sample after pre-stretch
(same definition for ε(t) and R0as in this paper) and Rc
and Λcare the compressed radius (i.e. the plate radius) and
compressed aspect ratio, respectively.
Rasmussen et al. (2010) have tried to improve the
analytical correction factor. They found an empirical
function for the correction factor, which ensures less than
3% deviation from their FiSER simulations. A Newtonian
sample was used in these simulations and the shear
correction factor was determined in the linear regime. No
pre-stretching was done, meaning that the simulations start
with cylindrical initial geometries. These geometries have
a compressed aspect ratio range of Λc=0.2 to 1.5. This
correction factor is rewritten in terms of strain and pre-strain
by Huang et al. (2012). The empirical correction factor is
(Rasmussen et al. 2010; Huang et al. 2012):
fshear =1+exp 5(ε(t ) +εpre )/3Λ3
c
3Λ2
c1
. (38)
The correction factors from literature do not include the
strain rate. This is because the theory of Spiegelberg et al.
(1996) is based on a Newtonian (viscous) fluid. In this
Fig. 8 Extensional viscosity of a filament extended at a strain rate of
˙ε=10 s1and a temperature of T=160 °C. Geometry 2 is used
with a compressed aspect ratio of Λc=0.5 (see Table 1). The numeri-
cal simulations are performed with the XPP model and the extensional
viscosity is found with Eq. 28. The pure uniaxial solution is found by
solving the XPP model for a pure uniaxial flow. The linear viscoelas-
tic envelope is calculated using Eq. 8. In all figures, a similar coloring
scheme is used for the various results
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type of fluids, the viscous stresses are linearly correlated
to the local strain rates and therefore drop out of the
analytical expression of the shear correction factor. Hence,
these correction factors are only valid for small strains in the
linear viscoelastic region (Spiegelberg et al. 1996). Since
FiSER experiments can be in the non-linear regime even
at small strains (as shown in Fig. 8), the correction factors
found for Newtonian fluids are revisited in the following
sections.
Geometry dependency
The non-linear shear correction factor is defined as the
ratio of the pure uniaxial viscosity (XPP) and the uniaxial
viscosity obtained from the FiSER simulation (as given
in Eq. 29). For both iPPs and LLDPE, a clear non-linear
behaviour is seen, since the simulated extensional viscosity
already deviates from the LVE at very small strains (see
Fig. 8a–c). The three materials show distinct differences in
rheological behaviour. In Fig. 8d, the extensional viscosity
of an arbitrary material, which will be indicated as non-
linear material, is presented. The extensional viscosity of
this material is tuned so that it shows more pronounced
non-linear behaviour at Hencky strains smaller than one as
compared to the three polymer melts.
The effect of the initial aspect ratio on the correction
factorisshowninFig.9. In this figure, it can be seen that
the correction factor deviates more from 1 with a decreasing
initial aspect ratio. This corresponds with the theory of
Spiegelberg, where it is stated that a larger correction for
shear contributions is needed for small initial aspect ratios
(Spiegelberg et al. 1996). To evaluate the models given in
literature, Eqs. 37 and 38 are calculated for the given initial
aspect ratios. From Fig. 9, it follows that the empirical
relation of Rasmussen et al. (2010) is better in predicting the
correction factor than the model of Spiegelberg et al. (1996).
These correction factors from literature are based on a
viscous (linear) Newtonian material. But the iPP used in this
study is non-linear viscoelastic. It is thus quite surprising
that the empirical shear correction factor of Rasmussen et al.
(2010) is quite accurate for the chosen initial aspect ratios.
Also note that a different reference is used in the shear
correction factor compared to the work of Rasmussen et al.
(2010), see Eq. 29. It is found that the pure uniaxial solution
of the non-linear constitutive model ¯η+
XPP should be used
instead of the linear viscoelastic viscosity ¯η+
LV E , because
they do not match (see Fig. 8).
To define a more accurate correction factor based on the
simulated (non-linear) shear correction factor in Fig. 9,a
modification of the shear correction factors of Spiegelberg
et al. (1996) and Rasmussen et al. (2010) is proposed. This
equation is found by fitting the simulated correction factors
giveninFig.9with the following function:
fshear =1+exp(a(ε(t) +εpre )/3b)
3Λ2
c1
,(39)
with aand bthe fitting parameters. For the four compressed
aspect ratios, aturns out to be approximately 4, while b
decreased with increasing compressed aspect ratio. To fit
bper compressed aspect ratio, a function needs to be used
that decreases to zero for infinite compressed aspect ratios.
This because then the shape approaches a perfect cylinder
for which no correction is needed. Therefore, the following
exponential function is used to fit b:
b=exp(Λc/c). (40)
Fig. 9 Simulated shear correction factor compared with the shear cor-
rection factors found in literature for different initial aspect ratios. The
top figure shows the simulation results compared to the theoretical
relation of Spiegelberg et al. (1996) and (Nielsen et al. 2008)givenin
Eq. 37. The bottom figure shows the simulation results compared to the
empirical relation of Rasmussen et al. (2010) and (Huang et al. 2012)
giveninEq.38.TheiPP
2(characterized by Grosso et al. (2019)) is
extended with a strain rate of ˙ε=10 s1at a temperature of T=160
°C
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Good agreement with bper compressed aspect ratio is
found for c=1. Combining Eqs. 39 and 40 with a=4
and c=1 results in the following proposed non-linear shear
correction factor:
fshear =1+exp(4(ε(t ) +εpre)/3exp(Λc))
3Λ2
c1
.
(41)
In Fig. 10, this proposed shear correction factor is
compared with the simulated shear correction factors. It
can be seen that the error of the improved correction factor
relation is within 3% for all investigated compressed aspect
ratios. Because of the choice of using pre-defined free
surface shapes (circular and ellipsoidal), it is possible to
rewrite Eq. 41 in terms of length L0and radius R0after
pre-strain using Eqs. 2and 3.
Because the proposed shear correction factor in Eq. 41
does not depend on the constitutive material behaviour,
this correction can be done without a-priori knowledge on
the rheological material behaviour. The measured viscosity
should be corrected with the shear correction factor given in
Eq. 41, i.e.:
¯η+
corr η+
Rfshear,(42)
where ¯η+
corr is the corrected extensional viscosity and ¯η+
Rthe
measured viscosity which is found from the measured force
on the plate and the mid-radius evolution using Eq. 28.
Rate independency
It is known that the non-linear behaviour of polymers is
rate dependent. Therefore, the rate dependency of the shear
correction factor is investigated. In the FiSER, the radius
of the sample is controlled in order to produce a locally
purely uniaxial flow. Hence, the strain as a function of time
is found from ε(t) εt. The modified shear correction
factor in Eq. 41 thus only depends on the strain rate via
the strain. Hence, when plotting the shear correction factor
versus strain, no difference between shear correction factors
at different strain rates is expected, because no other strain
Fig. 10 Simulated correction factor compared with the modified
relation for the shear correction factor given in Eq. 41 for different
compressed aspect ratios. The iPP2(characterized by Grosso et al.
(2019)) is extended with a strain rate of ˙ε=10 s1at a temperature
of T=160 °C
rate dependency is present in the modified shear correction
factor. In the simulations, the minimum applied strain rate
is chosen as ˙ε=0.71 s1. This is a typical strain
rate applied in filament stretching rheometers (Pepe et al.
2020). Anna et al. (2001) presented a detailed discussion
of gravitational sagging. Herein, they show that sagging
becomes significant when capillary forces in the neck near
the axial mid-plane are no longer able to overcome the
axial body force. This leads to a critical strain rate ˙εsag
ρgL00, which must be exceeded in order to minimize
the role of gravitational sagging. For the materials and
geometries used in this study, the critical strain rate is
on the order of 0.001 to 0.01 s1. Therefore gravitational
sagging plays no role in the following simulation results.The
maximum applied strain rate in the FiSER simulations is
10 s1. This is done to investigate at least one decade of
strain rates. Again, geometry 2 is used with a compressed
aspect ratio of Λc=0.5. Only the results of iPP2(Grosso
et al. 2019) are presented here, but it should be noted
that the same conclusions about the rate dependency apply
to the other compressed aspect ratios and the other two
polymer melts. The simulated shear correction factors at
different strain rates for iPP2are shown in Fig. 11.At
a compressed aspect ratio of Λc=0.5, the strain rate
only slightly (2%) affects the shear correction factors.
Therefore, it is concluded that the correction factor is strain
rate independent for the three polymer melts tested so
far. Simulations in a range of compressed aspect ratios
and strain rates are performed to verify the strain rate
independency.
Material independency
According to the correspondence principle of linear
viscoelasticity, the same shear correction factor should be
valid for all types of fluids (Spiegelberg et al. 1996).
However, in case of non-linear viscoelastic materials, this
is not necessarily true, because initial flow conditions
affect chain conformation at all later times (Spiegelberg
et al. 1996). Therefore, the shear correction factors of
the iPP1of Roozemond et al. (2015) and the LLDPE are
shown in Fig. 12. From this figure, it follows that these
materials have approximately the same shear correction
Fig. 11 Simulated correction factors for the iPP2of Grosso et al.
(2019) at strain rates of ˙ε=0.71 s1,˙ε=1.71 s1,˙ε=4.14 s1and
˙ε=10.0 s1using geometry 2 with Λc=0.5 (for T=160 °C)
704 Rheol Acta (2021) 60:691–709
Content courtesy of Springer Nature, terms of use apply. Rights reserved.
Fig. 12 Simulated correction factors for LLDPE, iPP1and the non-
linear material compared to the modified shear correction factor given
in Eq. 41. The filaments are extended with a strain rate of ˙ε=10 s1
at a temperature of T=160 °C
factor as the iPP2of Grosso et al. (2019). This means that
although the materials exhibit completely different non-
linear rheological behaviour, the shear correction factors
match. This supports the proposed shear correction factor,
Eq. 41, which appears to be independent of the type of
polymer and independent of the strain rate.
Although the polymers have distinctive rheological
behaviours, it follows from Fig. 7that for a Hencky strain
smaller than 1 the shape of the geometries of the iPP
and the LLDPE is almost the same. Correspondingly, the
effective strain rate distributions over the mid-radius are
similar for the three polymer melts, as shown in Fig. 13.
The distributions are plotted at Hencky strains of ε=0.5,
ε=1, ε=2andε=3, where the shear contributions
decrease for increasing strains (see Fig. 12). So, although
the polymer melts do not follow the LVE at relatively
Fig. 13 Effective strain rate distributions for the iPPs, LLDPE and the
arbitrary non-linear material at strains of ε=0.5, ε=1, ε=2and
ε=3. The filaments have a compressed aspect ratio of Λc=0.5 and
are extended with a strain rate of ˙ε=10 s1at T=160 °C
small strains (i.e. show some non-linear behaviour at small
strains), the mid-radius-based controller ensures that the
strain rate distributions are approximately the same. Hence,
geometrical effects dominate the non-linear effects of these
materials.
Since the rheological behaviour of these materials is
different, the similar effective strain rate distributions still
result in different stress distributions, as shown in Fig. 14.
Herein, the trace of the average conformation tensor is used
as an indicator for stress. This stress value is normalized
by dividing it by its mean value to compare the shape
of the stress distributions. For increasing strain, the stress
difference between the middle of the sample and the free
surface increases. This is due to the distribution in strain rate
over the mid-radius. For the LLDPE a more homogeneous
stress distribution is observed at the highest strains (ε=3).
This is related to the strain hardening behaviour of this
material. Because the strain hardening starts at a strain of
2 for LLDPE, where the strain rate distribution is nearly
flat, the associated stress increase is nearly homogeneous
over the mid-radius, and therefore the normalized stress
distribution flattens. In case of the iPPs, there is no clear
strain hardening behaviour present for strains smaller than
3. Hence, the non-homogeneous flow at small strains and
the resulting stress distribution are still visible at larger
strains. The stress distributions of these iPPs eventually
flattens after strains of 3, because of the homogeneous strain
distribution, which results in a more homogeneous stress
increase at the mid-radius.
Despite the different stress distributions over the
midfilament radius for the various materials, Fig. 13 shows
that the actual strain rate corresponds to the applied strain
rate at a radial position r/R around 0.72. Similarly, at this
location, the stress corresponds to the average stress of the
relevant profile. Therefore, using the extensional viscosity
Fig. 14 Stress distributions for the iPPs, LLDPE and the arbitrary non-
linear material at strains of ε=0.5, ε=1, ε=2andε=3. Here, the
normalized trace of the average conformation tensor is given, which
gives an indication of the relative stress distributions. The compressed
aspect ratio is Λc=0.5 and the extension is performed at a strain rate
of ˙ε=10 s1at T=160 °C
705Rheol Acta (2021) 60:691–709
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at the applied strain rate provides a consistent reference in
the shear correction factor (Eq. 29) and causes its universal
character. This occurrence of a fixed location at which applied
and actual strain rate match, is similar to the behaviour of non-
linear materials in capillary flows (Sch¨ummer and Worthoff
1978). Also there, deviations from the material-independent
Sch¨ummer correction factor occur in case of strongly non-
linear behaviour (Macosko 1994).
So from Fig. 13, it follows that at Hencky strains larger
than 1, the strain rate distribution over the mid-radius
becomes flattened and converges to the applied strain rate
of ˙ε=10 s1. Hence, if the non-linear effects before this
strain do not significantly affect the geometry (as is the
case for these polymer melts), then strain hardening after
a strain of 1 would not affect the shear correction factor
significantly. In case that the non-linear effects are larger
at small strains, it is hypothesised that the shear correction
factor in Eq. 41 is no longer valid. Strain hardening would
then affect the shape of the geometry, because there is
a strain rate distribution of about 10% at a strain of 1,
regardless of the material.
To test this hypothesis, an artificial non-linear material
is introduced. The relaxation spectrum and non-linear
parameters are given in Table 3.InFig.8d, it can be seen
that a clear strain hardening at small strains is present for
this artificial non-linear material. A simulation with this
non-linear material at a strain rate of ˙ε=10 s1results
in a shear correction factor as shown in Fig. 12. Herein,
the simulated shear correction factor does not match our
proposed shear correction factor. Hence, the pronounced
strain hardening at small strains affects the geometry and
the strain rate and stress distributions over the mid-radius.
In other words, the non-linear material’s plate velocity
needs to be higher to follow the ideal mid-radius with
the controller in case of strain hardening. Note that for
increasing compressed aspect ratio the deviation of the shear
correction factor from the simulated shear correction factor
becomes smaller. So for highly non-linear materials it is
even more important to increase the initial aspect ratio, and
therefore reduce the amount of correction needed.
At strains larger than 1.5, it can be seen in Fig. 12 that
the shear correction factor is slightly larger than one. The
reason for this is the conformation history which has not
equilibrised yet. The history effects for the used strain rate
of 10 s1will equilibrise at strains larger than 3. In Fig. 15,
it can be seen that for a strain rate of ˙ε=1.71 s1this
equilibration takes place at relatively smaller strains. This is
because there is more time for the polymer to equilibrise.
Note that even for this highly non-linear material no strain
rate dependency is observed for small enough strains.
For a wide range of polymer melts and solutions, and
even more complex polymeric materials such as ionomers
Fig. 15 Shear correction factor of the arbitrary non-linear material for
strain rates of ˙ε=1.71 s1and ˙ε=10 s1. The compressed aspect
ratio is Λc=0.5 and the extension is performed at T=160 °C
and transient polymer networks pronounced deviations from
linear viscoelasticity (such as strain hardening) only start
to appear at strains larger than one (McKinley and Sridhar
2002; Costanzo et al. 2016; Shabbir et al. 2017; Arora et al.
2017;M¨unstedt 2018). Therefore, it is possible to use the
proposed shear correction factor in Eq. 41 in a wide range
of FiSER experiments. However, for specific materials, this
highly non-linear behaviour in extensional flow can be
observed at strains much smaller than one, as for instance
for gluten gels, supramolecular polymers and glass fiber–
filled polymers (Ng and McKinley 2008;F
´
erec et al. 2009;
Cui et al. 2018). For these cases, care should be taken since
the proposed shear correction factor for filament stretching
rheometry is no longer valid, as shown in the case of the
artificial non-linear material. However, for an increased
initial aspect ratio, the shear contributions decrease and
the flow becomes more homogeneous. Therefore, the strain
rate distributions over the mid-radius will be flattened
and converge to the applied strain rate at smaller strains.
This implies that a near locally homogeneous flow can
be achieved before the start of the strain hardening of
the non-linear material. As a result, the simulated shear
correction factor converges towards the proposed universal
shear correction factor, as shown in Fig. 12.
Conclusions
A complete numerical tool, reproducing the flow and
resulting forces (rheology) of two iPPs, a LLDPE and
an artificial non-linear material in a FiSER, is presented.
In particular, finite element simulations are used that
simultaneously solve the mass balance and the momentum
balance using a non-linear viscoelastic constitutive equation
(XPP model). Also, a controller is added to mimic
the locally controlled uniaxial extensional flow in the
FiSER. The simulations are validated by comparing
force results with experiments and analytical predictions.
The typical force overshoot at the start of a filament
stretching experiment, caused by shear flow near the no-slip
boundaries of the pistons, is also present in the simulations.
Therefore, it is possible to investigate the shear effects for
706 Rheol Acta (2021) 60:691–709
Content courtesy of Springer Nature, terms of use apply. Rights reserved.
non-linear viscoelastic materials. Existing shear correction
factors perform well for sufficiently high Λcvalues, even
in the non-linear viscoelastic regime. To correct for shear
effects at small Λcvalues, an empirical correlation for
the non-linear shear correction factor is presented. This
shear correction factor shows good agreement with the
simulations of the polymer melts and has been validated for
multiple materials at different strain rates and initial aspect
ratios. For a wide range of polymer melts and solutions, and
even more complex polymeric materials such as ionomers
and transient polymer networks, which show moderate non-
linear behaviour at strains smaller than one, it is possible
to use the universal shear correction factor proposed in this
study. However, in specific materials such as gluten gels,
supramolecular polymers and glass fiber–filled polymers
which show highly non-linear behaviour at strains smaller
than one, the universal shear correction factor should be
used with care. For these materials, it is important to use
an initial geometry which ensures a near homogeneous
purely uniaxial flow before the start of the highly non-
linear behaviour of the material. This can be achieved by
increasing the initial aspect ratio. In future research it is
interesting to investigate how the shear correction factor
depends on the constitutive behaviour of these highly non-
linear materials.
Acknowledgements The authors thank Dr. M.A. Hulsen at the Eind-
hoven University of Technology (TU/e), Eindhoven, the Netherlands
for providing access to the Toolkit for Finite Element Method (TFEM)
software libraries.
Declarations
Conflict of interest The authors declare no competing interests.
Open Access This article is licensed under a Creative Commons
Attribution 4.0 International License, which permits use, sharing,
adaptation, distribution and reproduction in any medium or format, as
long as you give appropriate credit to the original author(s) and the
source, provide a link to the Creative Commons licence, and indicate
if changes were made. The images or other third party material in this
article are included in the article’s Creative Commons licence, unless
indicated otherwise in a credit line to the material. If material is not
included in the article’s Creative Commons licence and your intended
use is not permitted by statutory regulation or exceeds the permitted
use, you will need to obtain permission directly from the copyright
holder. To view a copy of this licence, visit http://creativecommons.
org/licenses/by/4.0/.
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Being able to properly model the material structure formation during processing is a fundamental step to predict final product properties, especially for semicrystalline polymers, like isotactic PolyPropylene (iPP), which can develop a multiplicity of different crystalline phases and morphologies. For this reason, in the present work a complete model is presented which can predict the complex structure formation of iPP in conditions comparable to injection moulding. The model includes a full coupling between the non-isothermal flow of a non-linear viscoelastic fluid and its crystallization process, properly capturing the mutual interaction between the two and is implemented in a finite element framework and as such applicable for general processing applications. The model is the result of many years of numerical and experimental research in our group and finally provides a complete simulation tool able to reproduce the complex iPP crystallization behaviour in conditions equivalent to processing. The model can predict not only the local crystalline composition, distinguishing between the multiple phases and morphologies that can develop inside iPP, but also the effect of the structure formation on the rheology. Comparisons with the unique in-situ data of Troisi et al. [1] demonstrate the good performance of the model and encourage further research to adapt the model to simulate other relevant processes and processing conditions. The results presented here are input for future work on structure related mechanical properties, see for example Caelers et al. [2,3]. Notice that the approach as presented here is not specific for iPP. A similar methodology, sometimes with additional modelling, is used for other polymers. 1
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We present unique nonlinear shear and extensional rheology data of unentangled amorphous polyester ionomers based on polyethers and sulphonated phthalates with sodium/lithium counterions. Previous linear viscoelastic measurements showed significant elasticity in these ionomers due to the formation of strong ionic aggregates. These ionomer melts exhibit viscoelastic properties similar to well-entangled melts with an extended rubbery plateau. To evaluate the effects of nonlinear deformation, the rheology of these ionomers was investigated using uniaxial extension and shear. The measurements were performed on a filament stretching rheometer and on a strain controlled rotational rheometer equipped with a cone-partitioned-plate setup. In extension, ionomer samples exhibited a decreasing strain hardening trend with increasing extension rates. At the same Weissenberg number, the same strain hardening was observed for different counterions. The presence of high solvating poly(ethylene oxide), PEO, along the backbone in the coionomer with poly(tetramethylene glycol), PTMO, increases the maximum Hencky strain at fracture, thus adding ductility to the brittle PTMO-Na ionomer. As a result, the coionomer deforms much more compared to PTMO-Na, but eventually, both fracture. On the other hand, whereas PTMO-Na cannot be sheared due to wall slip, the coionomer deforms in shear and eventually suffers from edge fracture instabilities. From the above, a picture emerges suggesting that PEO coionomers enhance ductility, make fracture smoother and offer a compromise of mechanical performance and ion conduction.
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We investigate the nonlinear shear and uniaxial extensional rheology of entangled polystyrene (PS) melts and solutions having the same number Z of entanglements, hence identical linear viscoelasticity. While experiments in extensional flows confirm that PS melts and solutions with the same Z behave differently, respective transient and steady data in simple shear over the largest possible range of rheometric shear rates (corresponding to Rouse-Weissenberg numbers from 0.01 to 40) demonstrate that melts and solutions exhibit identical behavior. Whereas the differences between melts and solutions in elongational flows are due to alignment-induced friction reduction (more effective in melts than in solutions), in shear flows they disappear since the rotational component reduces monomeric alignment substantially. Recent molecular dynamics simulations of entangled polymers show that rotation induces molecular tumbling at high shear rates, and here a tube-based model involving tumbling effects is proposed in order to describe the response in shear. The main outcome is that tumbling can explain transient stress undershoot (following the overshoot) at high shear rates. Hence, the combination of tumbling in shear and friction reduction in extension successfully describes the whole range of experimental data and provides the basic ingredient for the development of molecular constitutive equations.
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The primary and secondary crystallization kinetics of a homogeneous linear low-density polyethylene were characterized as function of cooling rate, pressure and flow strength. Our approach to describe primary crystallization is based on nucleation and growth of spherulites, quantified well below the melting temperature using small-angle light scattering. The description of the two-step secondary process is coupled to primary crystallization using a convolution integral, for which the parameters were determined from (fast-) differential scanning calorimetry. Extended-dilatometry was used to investigate the effect of different thermomechanical histories. Parameters were determined for an existing model that couples molecular stretch to both nucleation rate and fibrillar growth rate. Excellent agreement is shown between calculated and experimentally obtained crystallization kinetics in conditions representative for those found in real-life processing conditions. This opens the possibility to calculate in detail the evolution of and the final crystallinity structure in products such as blown film or extruded tape.